Applications of homotopy theory to 4D geometry, number theory, and physics

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DMS-0406461 Jack MoravaHomotopy theory plays an important role in recent work in four-dimensional geometry, physics, and number theory: it is a technically powerful ideology, which exposes deep connections among topics which may on the surface seem unrelated. Recent work of Madsen, Tillmann, Weiss, Cohen, and others on the stable cohomology of the moduli space of Riemann surfaces is based on the study of certain cobordism categories suggested by ideas from string physics. These constructions have four-dimensional analogs (of similar significance in physics), which display surprising connections to the pseudoisotopy theory developed in the 80's and 90's by Hatcher, Waldhausen, and others. That theory involves the algebraic K-theory of the integers in a nontrivial way; and the latter subject is now seen to play an important role in an apparently unrelated circle of ideas connecting the absolute Galois group of the rational number field to the theory of motives in algebraic geometry (through work of Deligne, Goncharov, Kontsevich, and others). These ideas are in turn involved in work of Connes, Kreimer, and others on a reinterpretation of the classical theory of renormalization in physics. One of the central notions of this proposal is the hope of finding in differential topology an analog of the algebraic geometers' mixed Tate motives, which would be related to the algebraic K-theory of the integers as that subject is to Waldhausen's algebraic K-theory of spaces.In less technical terms: homotopy theory provides a body of techniques for studying mathematical objects and their deformations on an equal footing; indeed, it is equally happy studying deformations of deformations, and so on. This proposal is concerned ultimately with two sets of ideas with roots in physics, one coming from modern string theory, the other from the more classical theory of renormalization. The former set of ideas has deep connections with differential topology, and the latter is related to recent developments in algebraic geometry and number theory. On the surface, differential topology and arithmetic algebraic geometry are far apart, but they are linked through the algebraic K-theory of the integers, which is at base a part of homotopy theory. [Because that subject deals so systematically with deformations of one theory into another, it provides a very convenient set of tools for relating such disparate subjects.] This proposal suggests a way of clarifying these linkages, based on ideas about the K-theory of spaces rather than numbers, which were first proposed by Waldhausen, and which have lately given further currency by workers in arithmetic geometry such as Soul\'e. This program would bring together these important recent developments in mathematics and physics in a conceptually satisfying way.